To find \(\cos(x)\) when \(\sin(x) = -\frac{1}{3}\), we can use the Pythagorean identity:

\[
\sin^2(x) + \cos^2(x) = 1
\]

### Step-by-Step Solution:

1. **Substitute the given value of \(\sin(x)\) into the identity**:
   \[
   \left(-\frac{1}{3}\right)^2 + \cos^2(x) = 1
   \]

2. **Simplify**:
   \[
   \frac{1}{9} + \cos^2(x) = 1
   \]

3. **Isolate \(\cos^2(x)\)**:
   \[
   \cos^2(x) = 1 - \frac{1}{9}
   \]

4. **Simplify further**:
   \[
   \cos^2(x) = \frac{9}{9} - \frac{1}{9} = \frac{8}{9}
   \]

5. **Take the square root** to find \(\cos(x)\):
   \[
   \cos(x) = \pm \sqrt{\frac{8}{9}} = \pm \frac{\sqrt{8}}{3} = \pm \frac{2\sqrt{2}}{3}
   \]

### Final Answer:

\[
\cos(x) = \pm \frac{2\sqrt{2}}{3}
\]

The sign depends on the quadrant in which the angle \(x\) lies.

---

Would you like further details or do you have any questions? Here are some related questions:

1. How do we determine the sign of \(\cos(x)\) given a specific interval for \(x\)?
2. What is the range of values for \(\sin(x)\) and \(\cos(x)\)?
3. How do you use the unit circle to find \(\sin(x)\) and \(\cos(x)\)?
4. What are the Pythagorean identities and how are they derived?
5. How do we find the exact value of \(\tan(x)\) when \(\sin(x)\) is known?

**Tip:** The sign of trigonometric functions depends on the quadrant where the angle lies, which can be determined using the ASTC (All Students Take Calculus) rule.

Discover how to find cos(x) when sin(x) = -1/3 with a detailed step-by-step solution using the Pythagorean identity. Understand the implications of the solution based on the quadrant in which the angle x lies. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation